Flat Vector Bundles over Parallelizable Manifolds

Examples of non-flat bundles of rank one

In this article we give several examples of line bundles on certain non-compact surfaces that cannot be equipped with a flat connection.

On algebraic Chern classes of flat vector bundles

We show that under some assumptions on the monodromy group some combinations of higher Chern classes of flat vector bundles are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch's conjecture on Chern classes of flat vector bundles on smooth complex projective varities but in some cases they give a more precise information. We also study Higgs version of Bloch's conjecture and analogous problems in the positive characteristic case.

向量丛的弱正性与弱平坦性

Weak positivity and weak flatness of vector bundles

本文研究诺特环上拟射影概形向量丛的弱正性，证明其在多种运算下保持，引入弱平坦丛并建立相关定理，最终用于刻画阿贝尔簇的商。

Characteristic classes of flat bundles and determinant of the Gauss-Manin connection

The purpose of this note is to give a survey on recent progress on characteristic classes of flat bundles, and how they behave in a family.

Chern invariants of some flat bundles in the arithmetic Deligne cohomology

In this note, we investigate the Chern classes of flat bundles in the arithmetic Deligne Cohomology, introduced by Green-Griffiths, Asakura-Saito. We show nontriviality of the Chern classes in some cases and the proof also indicates that generic flat bundles (in its moduli space) can be expected to have nontrivial classes, which is in contrast with some Gauss-Manin systems.

Unipotent flat bundles and Higgs bundles over compact Kaehler manifolds

We characterize those unipotent representations of the fundamental group $π_1(X,x)$ of a compact Kaehler manifold $X$, which correspond to a Higgs bundle whose underlying Higgs field is equal to zero. The characterization is parallel to the one that R. Hain gave of those unipotent representations of $π_1(X,x)$ that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle. We see that Hain's result and ours follow from a careful study of Simpson's correspondence between Higgs bundles and local systems.

Ricci-flat Deformations of Holomorphic Vector Bundles

In this paper we give a criterion for a deformation of a hermitian vector bundle to be Ricci-flat. As an application we show that on a Kähler manifold, every deformation of a vector bundle can be made Ricci-flat whereas on some Hopf manifolds, the non-existence of a Ricci-flat deformation is related to non-trivial vector bundles on the universal cover ${\mathbb C}^n\setminus\{0\}$. On a surface with $b_1(X)\not=0$ filtrable Ricci-rigid vector bundles prove to be very special. We apply this to Inoue and Hopf surfaces.

Varieties in positive characteristic with numerically flat log cotangent bundle

In this paper, we prove that a smooth projective globally $F$-split variety with numerically flat tangent bundle is an étale quotient of an ordinary abelian variety. We also show its logarithmic analog, which contains a characterization of toric varieties. We further prove that, without assumption of global $F$-splitting, a smooth projective separably rationally connected variety of arbitrary characteristic with numerically flat log cotangent bundle is a toric variety.

Local structure of Brill-Noether strata in the moduli space of flat stable bundles

We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact Kahler manifold, proving that these strata have quadratic algebraic singularities.

曲面平坦丛的上同调有界性及其应用

Cohomological boundedness for flat bundles on surfaces and applications

本文证明曲面及任意维数下平坦丛的上同调维数、Lefschetz识别及转向轨迹度数的普遍有界性，并引入部分差异运算以给出特征循环的闭式公式。

正特征下平向量丛的完全旗

Complete flags on flat vector bundles in positive characteristic

本文研究正特征下平向量丛的相关问题，证明亏格g≤1等价于X上任意平向量丛存在完全旗，并探索该结果的更广泛推广。

非零度情形下Higgs丛模空间的半平坦度量

Semi-flat metrics of the moduli spaces of Higgs bundles in the non-zero degree case

本文研究非零度情形下谱曲线光滑的Higgs丛，定义Hitchin纤维化的可积联络，引入半平坦度量并比较其与Hitchin度量的渐近行为。

卡拉比-丘流形上数值平坦希格斯丛与曲线半稳定希格斯丛的滤过

Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi-Yau manifolds

研究卡拉比-丘流形上的希格斯丛，证明数值平坦希格斯丛存在商为稳定局部自由H-nflat的Jordan-Hölder滤过，进而推出单连通卡拉比-丘流形上曲线半稳定希格斯丛判别式为零。

Projectively flat log smooth pairs

In this article, we study projective log smooth pairs with numerically flat normalized logarithmic tangent bundle. Generalizing works of Jahnke-Radloff and Greb-Kebekus-Peternell, we show that, passing to an appropriate finite cover and up to isomorphism, these are the projective spaces or the log smooth pairs with numerically flat logarithmic tangent bundles blown-up at finitely many points away from the boundary. On the other hand, the structure of log smooth pairs with numerically flat logarithmic tangent bundle is well understood: they are toric fiber bundles over finite étale quotients of…

数值平坦 Higgs 丛的消失 Chern 类

Vanishing Chern classes for numerically flat Higgs bundles

本文研究了 Higgs 丛的 ample 性，证明了 H-ample 和数值有效 Higgs 丛 Chern 类的非负性，并据此确立了数值平坦 Higgs 丛 Chern 类的消失性。

射影平坦叶状结构

Projectively flat foliations

本文研究了复射影流形上具有数值射影平坦切丛的正则余维1叶状结构，描述了其结构，并证明了其法丛伪有效或余法丛nef。

Toric flat families, valuations, and applications to projectivized toric vector bundles

Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.

关于曲线半稳定和数值平坦Higgs丛的概述

An overview on curve semistable and numerically flat Higgs bundles

本文通过Higgs-Grassmannian方案定义数值平坦等正性条件，证明满足条件的Higgs丛的性质，并探讨其与曲线半稳定Higgs丛的关系。

On deep Frobenius descent and flat bundles

Let R be an integral domain of finite type over Z and let f:X --> Spec R be a smooth projective morphism of relative dimension d >= 1. We investigate, for a vector bundle E on the total space X, under what arithmetical properties of a sequence (p_n, e_n)_{n \in \NN}, consisting of closed points p_n in Spec R and Frobenius descent data E_{p_n} \cong F^{e_n}^*(F) on the closed fibers X_{p_n}, the bundle E_0 on the generic fiber X_0 is semistable.

非常一般曲线上的平坦向量丛与非阿贝尔Hodge领域的码维

Flat Vector Bundles on Very General Curves and Codimension of Non-Abelian Hodge Loci

本文基于复Hodge结构的变体秩和水平，界定了非阿贝尔Hodge领域在相对de Rham模空间上的码维，并通过Landesman和Litt的成果，研究了等monodromy叶状结构及代数monodromy群的李代数秩。

A Riemann-Roch theorem for flat bundles, with values in the algebraic Chern-Simons theory

Let $f: X \to S$ be flat morphism over an algebraically closed field $k$ with a relative normal crossings divisor $Y\subset X$, $(E, \nabla)$ be a bundle with a connection with log poles along $Y$ and curvature with values in $f^*Ω^2_{k(S)}$. Then the Gauß-Manin sheaf $R^if_*(Ω^*_{X/S}({\rm log} Y)\otimes E)$ carries a Gauß-Manin connection $GM^i(\nabla)$. We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of $\nabla$, $GM^i(\nabla)$ and the top Chern class of $Ω^1_{X/S}({\rm log}Y)$.

Cleanliness and log-characteristic cycles for vector bundles with flat connections

Let $X$ be a proper smooth algebraic variety over a field $k$ of characteristic zero and let $D$ be a divisor with simple normal crossings. Let $M$ be a vector bundle over $X-D$ equipped with a flat connection with possible irregular singularities along $D$. We define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of $D$. When this condition is satisfied, we compute explicitly the associated log-characteristic cycle, and relate it to the so-called refined irregularities. As a corollary of a log-varia…

Punctual characterization of the unitary flat bundle of weight 1 PVHS and application to families of curves

In this paper we consider the problem of pointwise determining the fibres of the flat unitary subbundle of a PVHS of weight one. Starting from the associated Higgs field, and assuming the base has dimension $1$, we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of …

On the complement of a hypersurface with flat normal bundle which corresponds to a semipositive line bundle

We investigate the complex analytic structure of the complement of a non-singular hypersurface with unitary flat normal bundle when the corresponding line bundle admits a Hermitian metric with semipositive curvature.

Hermitian-Einstein connections on principal bundles over flat affine manifolds

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal $G$-bundle $E_G$ over $M$ admits a Hermitian-Einstein structure if and only if $E_G$ is polystable. A polystable flat principal $G$--bundle over $M$ admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for …

A remark on an algebraic Riemann-Roch formula for flat bundles

We propose an algebraic Riemann-Roch formula for moving flat bundles on contant families in characteristic zero with values in the ring of algebraic differential characters. This formula lifts Grothendieck-Riemann-Roch formula in the Chow groups and Bismut-Lott formula in the odd dimensional Betti cohomology with $\C/\Z$ coefficients over the field of complex numbers.

Curvature restrictions on a manifold with a flat Higgs bundle

We shall prove a semi-negative curvature property for a manifold with a flat admissible Higgs bundle.

Definable structures on flat bundles

A flat vector bundle on an algebraic variety supports two natural definable structures given by the flat and algebraic coordinates. In this note we show these two structures coincide, subject to a condition on the local monodromy at infinity which is satisfied for all flat bundles underlying variations of Hodge structures.

Flat parabolic vector bundles on elliptic curves

We describe the moduli space of logarithmic rank 2 connections on elliptic curves with 2 poles.

Higgs reductions and numerically flat principal Higgs bundles

I consider principal Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced by Bruzzo and Graña Otero. I prove that a principal Higgs bundle $\mathfrak{E}=(E,φ)$ is H-nflat is either stable or there exists a Higgs reduction of $\mathfrak{E}$ to a parabolic subgroup $P$ of $G$ such that the principal $L$-bundle $\mathfrak{E}_L$ obtained by extending the reduced Higgs bundle $\mathfrak{E}_P$ to the Levi factor $L$ is H-nflat and stable; and as consequence, $H^{*}(\mathfrak{E},\mathbb{R})$ is isomorphic to the cohomology ring of the associated graded object $\ma…

Higher codimensional Ueda theory for a compact submanifold with unitary flat normal bundle

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher-codimensional generalization of Ueda's result, we give a sufficient condition for the existence of a non-singular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.

Simpson-Mochizuki Correspondence for $λ$-Flat Bundles

The notion of flat $λ$-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi--Hitchin-type theorem for $λ$-flat bundles ($λ\neq 0$), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable $λ$-flat bundles ($λ\neq 0$). On the other hand, we study two applications of the Simpson--Mochizuki correspond…

幂零希格斯丛与平坦联络族

Nilpotent Higgs bundles and families of flat connections

本文研究首项为幂零希格斯场的平坦联络族，证明其单值性与首项为正则希格斯丛的族相同，并由此得到和乐迹关于参数有理幂的渐近指数性质。

Descent of restricted flat Mittag-Leffler modules and generalized vector bundles

A basic question for any property of quasi--coherent sheaves on a scheme $X$ is whether the property is local, that is, it can be defined using any open affine covering of $X$. Locality follows from the descent of the corresponding module property: for (infinite dimensional) vector bundles and Drinfeld vector bundles, it was proved by Kaplansky's technique of dévissage already in \cite[II.\S3]{RG}. Since vector bundles coincide with $\aleph_0$-restricted Drinfeld vector bundles, a question arose in \cite{EGPT} of whether locality holds for $κ$-restricted Drinfeld vector bundles for each infini…

Vector bundles with numerically flat reduction on rigid analytic varieties and $p$-adic local systems

We show how to functorially attach continuous $p$-adic representations of the profinite fundamental group to vector bundles with numerically flat reduction on a proper rigid analytic variety over $\mathbb{C}_p$. This generalizes results by Deninger and Werner for vector bundles on smooth algebraic varieties. Our approach uses fundamental results on the pro-étale site of a rigid analytic variety introduced by Scholze. This enables us to get rid of the smoothness condition and to work in the analytic category. Moreover, under some mild conditions, the functor we construct gives a full embedding …

关于复结构与全纯向量丛对的形变分裂

On splittings of deformations of pairs of complex structures and holomorphic vector bundles

本文研究复流形与全纯向量丛对的Kuranishi空间分裂问题，证明Kähler情形和复可平行化nilmanifold平凡丛情形下可分解为直积，并给出非分裂的nilmanifold反例。

Regulators and characteristic classes of flat bundles

This is a substantial revision of the older version of this paper. The main result of the old version (the equality, up to a factor of 2 of the Beilinson and Borel regulators) is now a conjecture. The main results give equality of Beilinson chern classes and Cheeger-Simons-Chern classes in various situations such as for flat bundles over quasi projective varieties. We also prove the equality (up to a factor of two) of the Borel regulator element and the universal Cheeger-Simons Chern class.

Metrics on semistable and numerically effective Higgs bundles

We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, establishing semistability criteria …

Numerical flatness and principal bundles on Fujiki manifolds

Let $M$ be a compact connected Fujiki manifold, $G$ a semisimple affine algebraic group over $\mathbb C$ with one simple factor and $P$ a fixed proper parabolic subgroup of $G$. For a holomorphic principal $G$--bundle $E_G$ over $M$, let ${\mathcal E}_P$ be the holomorphic principal $P$-bundle $E_G\rightarrow E_G/P$ given by the quotient map. We prove that the following three statements are equivalent: (1) ${\rm ad}(E_G)$ is numerically flat, (2) the holomorphic line bundle $\bigwedge^{\rm top} {\rm ad}({\mathcal E}_P)^*$ is nef, and (3) for every reduced irreducible compact complex analytic s…

Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces

Using Szenes formula for multiple Bernoulli series we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also certain multiple zeta values.

Irreducible flat SL(2,R)-connections on the trivial holomorphic bundle

We construct an irreducible holomorphic connection with SL(2,R)-monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in \cite{CDHL}.

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principa…

Kobayashi-Hitchin correspondence for tame harmonic bundles II

Let $X$ be a smooth projective complex variety with an ample line bundle $L$, and let $D$ be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on $X-D$ and $μ_L$-stable parabolic $λ$-flat bundles with trivial characteristic numbers on $(X,D)$. Especially, we obtain the quasiprojective version of the Corlette-Simpson correspondence between flat bundles and Higgs bundles.

Any flat bundle on a punctured disc has an oper structure

We prove that any flat G-bundle, where G is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in arXiv:math/0508382. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine Weyl groups on the homology of these deformations generalizing representations constructed by Lusztig.

Pseudo-effective and numerically flat reflexive sheaves

In this note, we discuss the concept of pseudoeffective vector bundle and also introduce pseudoeffective torsion-free sheaves over compact Kähler manifolds. We show that a pseudoeffective reflexive sheaf over a compact Kähler manifold with vanishing first Chern class is in fact a numerically flat vector bundle. A proof is obtained through a natural construction of positive currents representing the Segre classes of pseudoeffective vector bundles.

非阿贝尔 Hodge 对应与具有固定基本结构的 Sasakian 流形上平坦丛的模空间

Non-abelian Hodge correspondence and moduli spaces of flat bundles on Sasakian manifolds with fixed basic structures

本文研究紧 Sasakian 流形上简单平坦丛的模空间分解，并在模空间层面刻画非阿贝尔 Hodge 对应，进而得到 Hitchin 真泛性质的类比。

紧致Sasakian流形上的Higgs丛与平坦联络，II：拟正则丛

Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundles

本文在拟正则紧致Sasakian流形上引入基本向量丛的拟正则性与正则性概念，建立轨形基本群半单表示与Higgs轨形丛之间的非阿贝尔Hodge对应，并将其推广到平坦丛与基本Higgs丛，同时证明了数值平坦丛刻画的Sasakian类比。

Mall bundles and flat connections on Hopf manifolds

A Mall bundle on a Hopf manifold H is a holomorphic vector bundle whose pullback to the universal cover of H is trivial. We define resonant and non-resonant Mall bundles, generalizing the notion of the resonance in ODE, and prove that a non-resonant Mall bundle always admits a flat holomorphic connection. We use this observation to prove a version of Poincare-Dulac linearization theorem, showing that any non-resonant invertible holomorphic contraction of a complex space is linear in appropriate holomorphic coordinates. We define the notion of resonance in Hopf manifolds, and show that all non-…

Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles and complex variations of Hodge structure

Lecture notes from the Concentrated Graduate Course preceding the Workshop on Hodge Theory in String Theory at the Fields Institute in Toronto, November 11--15, 2013.

Moduli of flat bundles on open Kaehler manifolds

We consider the moduli space MN of flat unitary connections on an open Kaehler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection and L2 cohomologies with degenerating coefficients we construct a natural symplectic form F on MN. When U is quasi-projective we prove that F is actually a Kaehler form.

Lower bounds for Lyapunov exponents of flat bundles on curves

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non-expanding cusp monodromies. As an application we obtain the large genus limits of individual Lyapunov exponents in hyperelliptic strata of Abelian differentials. Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, …

Goldman form, flat connections and stable vector bundles

We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree $0$ over a compact Riemann surface and the affine bundle $\mathscr{A}\to\mathscr{N}$ of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of of the quasi-Fuchsian projective connections - local holomorphic sections of $\mathscr{A}$ - that allow to pull back the Liouville symplectic form on $T^{*}\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of the Goldman form to $\mathscr{A}$ by the Riemann-Hilbert correspondence coincides with …

数值平坦叶状结构与全纯泊松几何

Numerically flat foliations and holomorphic Poisson geometry

本文研究了紧Kähler流形上具有数值平坦切丛的光滑全纯叶状结构，证明了其诱导切丛分解等性质，并由此导出了射影流形上秩为2的Poisson结构的全局Weinstein分裂定理及零点存在性的新Hodge理论条件。

A theory of minimal K-types for flat G-bundles

The theory of minimal K-types for p-adic reductive groups was developed in part to classify irreducible admissible representations with wild ramification. An important observation was that minimal K-types associated to such representations correspond to fundamental strata. These latter objects are triples (x, r, beta), where x is a point in the Bruhat-Tits building of the reductive group G, r is a nonnegative real number, and beta is a semistable functional on the degree r associated graded piece of the Moy-Prasad filtration corresponding to x. Recent work on the wild ramification case of the …